I know the difference between the forward (explicit) Euler method and backward (implicit) Euler method in the context of approaching solutions to ODEs. However, I have a doubt regarding the following method: y_(n+1)=y_n+h/2(f(t_n,y_n)+f(t_(n+1),y_(n+1))) I am not sure whether it is an implicit or an explicit method. I tend to say it is implicit one step methode because we have to solve an equation to calculate yn+1.

vagnhestagn

vagnhestagn

Answered question

2022-09-07

I know the difference between the forward (explicit) Euler method and backward (implicit) Euler method in the context of approaching solutions to ODEs.
However, I have a doubt regarding the following method:
y n + 1 = y n + h 2 ( f ( t n , y n ) + f ( t n + 1 , y n + 1 ) )
I am not sure whether it is an implicit or an explicit method.
I tend to say it is implicit one step methode because we have to solve an equation to calculate y n + 1 .
Thank you for your help.

Answer & Explanation

Jase Powell

Jase Powell

Beginner2022-09-08Added 11 answers

You're quite right.
An explicit method is given by y n + 1 = F ( t n , y n ), with no mention of y n + 1 on the RHS, and you can compute y n + 1 directly.
An implicit method is given by F ( t n , y n , y n + 1 ) = 0, and you have to solve an equation to find y n + 1 .
Both these definitions are for one-step methods: y n + 1 depends directly or indirectly just on t n and y n , not on any y k for k < n.

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